Ensuring Long-Term Safety in Stochastic Control Systems through Myopically Verifiable Probabilistic Certificates

Core Concepts

The paper proposes a novel technique called "probabilistic invariance" to efficiently design myopic controllers that can ensure long-term safety in stochastic control systems. This technique allows the controller to directly optimize for the probability of long-term safety or convergence, rather than just the short-term state constraints.

Abstract

The paper addresses the challenge of ensuring long-term safety in stochastic control systems, where traditional set-invariance based methods may not be sufficient due to the accumulation of uncertainties over time. To overcome this, the authors introduce a novel "probabilistic invariance" technique that characterizes the invariance conditions of the probability of interest, rather than the state space.
The key insights are:

An infinitesimal future value of a long-term probability provides explicit computable information about invariant sets in the probability space, unlike the state space.
This allows the authors to derive myopic conditions that can ensure long-term probabilistic safety or convergence.

The paper then integrates this technique into safe control and learning methods:

For control, the proposed technique can equip nominal controllers (e.g., neural networks) with long-term safety guarantees using either an additive modification or a constrained optimization approach.
For learning, the technique can be used to ensure long-term safety of the control policies during and after training.
The performance of the proposed techniques is demonstrated through numerical simulations.

Stats

None.

Quotes

"The key insight is that its probabilistic analog, namely an infinitesimal future value of a long-term probability, does provide explicit computable information about invariant sets in the space of probability values."
"Condition (39) is a forward invariance condition on probability, while typical control barrier function (CBF) based methods perform forward invariance on state space."

Key Insights Distilled From

Myopically Verifiable Probabilistic Certificates for Safe Control and Learning

by Zhuoyuan Wan... at arxiv.org 04-29-2024

https://arxiv.org/pdf/2404.16883.pdf

Deeper Inquiries

How can the proposed probabilistic invariance technique be extended to handle partially observable or time-varying systems

The proposed probabilistic invariance technique can be extended to handle partially observable systems by incorporating techniques from Bayesian inference. In partially observable systems, the state of the system is not directly observable, leading to uncertainty in the system's evolution. By integrating Bayesian methods, such as particle filters or Kalman filters, the probabilistic invariance technique can account for this uncertainty by updating the probability distribution over the system's state based on observations. This updated distribution can then be used to calculate the long-term safe probability, taking into consideration the partial observability of the system.
For time-varying systems, the probabilistic invariance technique can be adapted by introducing time-varying constraints and probabilities. By dynamically adjusting the constraints and probabilities based on the changing nature of the system over time, the technique can ensure long-term safety in systems with varying dynamics. This adaptation may involve updating the control policies or modifying the safety conditions to account for the time-varying nature of the system. Additionally, incorporating adaptive control strategies that adjust in real-time based on the system's behavior can enhance the applicability of the probabilistic invariance technique to time-varying systems.

What are the potential limitations or drawbacks of the additive modification and constrained optimization approaches for safe control

Limitations of Additive Modification Approach:

Sensitivity to Parameter Selection: The effectiveness of the additive modification approach heavily relies on the selection of the parameter κ. Inaccurate or suboptimal choices of κ may lead to subpar performance in ensuring long-term safety.

Complexity in Tuning: Tuning the parameter κ to achieve the desired balance between performance and safety can be a challenging task, requiring extensive experimentation and fine-tuning.

Drawbacks of Constrained Optimization Approach:

Computational Complexity: The constrained optimization approach may involve solving complex optimization problems, especially in real-time applications, leading to increased computational burden and potential delays in decision-making.

Limited Flexibility: The constraints imposed in the optimization problem may restrict the range of feasible solutions, potentially limiting the adaptability of the control policies to dynamic or uncertain environments.

Risk of Overfitting: Overfitting the control policies to the specific conditions of the training data may result in reduced generalization and robustness in real-world scenarios.

Can the probabilistic invariance framework be applied to other domains beyond control systems, such as decision-making under uncertainty or risk-aware planning

The probabilistic invariance framework can be applied to various domains beyond control systems, including decision-making under uncertainty and risk-aware planning. By extending the principles of probabilistic invariance to these domains, one can enhance the decision-making processes in complex and uncertain environments. Here are some potential applications:

Finance and Investment: In financial decision-making, the probabilistic invariance framework can be utilized to assess the long-term risks and uncertainties associated with investment strategies. By incorporating probabilistic safety constraints, investors can make more informed and risk-aware decisions.

Healthcare: In healthcare settings, the framework can aid in optimizing treatment plans by considering long-term safety probabilities and uncertainties. It can help healthcare providers navigate complex treatment options while ensuring patient safety and well-being.

Supply Chain Management: Applying probabilistic invariance to supply chain management can improve risk mitigation strategies and enhance decision-making processes in dynamic and uncertain supply chain environments. By considering long-term safety probabilities, organizations can better manage disruptions and uncertainties in the supply chain.

By adapting the probabilistic invariance framework to these diverse domains, stakeholders can make more robust and informed decisions in the face of uncertainty and risk.